A capacity approach to box and packing dimensions of projections and other images

Kenneth John Falconer*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Abstract

Dimension profiles were introduced by Falconer and Howroyd to provide formulae for the box-counting and packing dimensions of the orthogonal projections of a set E or a measure on Euclidean space onto almost all m-dimensional subspaces. The original definitions of dimension profiles are somewhat awkward and not easy to work with. Here we rework this theory with an alternative definition of dimension profiles in terms of capacities of E with respect to certain kernels, and this leads to the box-counting dimensions of projections and other images of sets relatively easily. We also discuss other uses of the profiles, such as the information they give on exceptional sets of projections and dimensions of images under certain stochastic processes. We end by relating this approach to packing dimension.
Original languageEnglish
Title of host publicationAnalysis, Probability and Mathematical Physics on Fractals
EditorsPatricia Alonso Ruiz, Joe P Chen, Luke G Rogers, Robert S Strichartz, Alexander Teplyaev
Place of PublicationSingapore
PublisherWorld Scientific Publishing
Number of pages14
ISBN (Electronic)9789811215537, 9789811215544
ISBN (Print)9789811215520
DOIs
Publication statusPublished - Mar 2020
Event6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals - Cornell, United States
Duration: 13 Jun 201717 Jun 2017
Conference number: 6
http://pi.math.cornell.edu/~fractals/

Publication series

NameFractals and Dynamics in Mathematics, Science and the Arts: Theory and Applications
PublisherWorld Scientific
Volume5
ISSN (Print)2382-6320

Conference

Conference6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals
Abbreviated titleFractals
Country/TerritoryUnited States
CityCornell
Period13/06/1717/06/17
Internet address

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